A280225 G.f.: Product_{k>=1} (1 + 3*x^(k^2)) / (1-x^k).

1, 4, 5, 9, 17, 34, 47, 75, 109, 165, 240, 341, 473, 671, 936, 1268, 1722, 2325, 3091, 4099, 5403, 7083, 9207, 11923, 15339, 19682, 25134, 31909, 40378, 50954, 64068, 80171, 100089, 124506, 154465, 191043, 235636, 289816, 355673, 435285, 531486, 647478

Offset: 0

Author

Vaclav Kotesovec, Dec 29 2016

Keywords

• nonn

Comments

Convolution of A279368 and A000041.

In general, if m >= 0 and g.f. = Product_{k>=1} (1 + m*x^(k^2)) / (1-x^k), then a(n) ~ exp(Pi*sqrt((2*n)/3) + 3^(1/4)*c*n^(1/4)/ 2^(3/4) - 3*c^2/(32*Pi)) / (4*sqrt(3)*sqrt(m+1)*n), where c = -PolyLog(3/2, -m).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000041, A033461, A279368, A280204, A280224.

Programs

Mathematica

nmax=50; CoefficientList[Series[Product[(1+3*x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt((2*n)/3) + 3^(1/4)*c*n^(1/4)/ 2^(3/4) - 3*c^2/(32*Pi)) / (8*sqrt(3)*n), where c = -PolyLog(3/2, -3).